The bicycle is a complex system to analyze. In Part 2, we talked a little about modeling the idealized passive rider-bicycle system, simplifying many things through assumptions but still capturing enough detail to go ahead with a reasonable analysis. The full analysis, particularly that involving the derivation of the equations of motion, is beyond the scope of this blog.

But to put it in simple words, what the analysis yields are two coupled second order, non-linear, differential equations in lean and steer. Then, these equations are linearized through a carefully followed algorithm to give us an eigenvalue problem. The eigenvalues gained from the characteristic equation help us assess the stability of the modes of bicycle motion. Eigenvalues are the cool numbers that give us an idea of the stability of the engineered system when the system is disturbed.

Fig 3 : What eigenvalues tell us about stability. To make an engineered system stable, we're all interested in attaining negative real numbers for eigenvalues. Stable motion of a bicycle has negative, real, eigenvalues. Courtesy : University of Michigan, Dynamics And Controls

Many years of research has allowed us to understand the nature of the bicycle's linearized equation of motion (LEOM). The LEOM, expressed in terms of small changes in the lateral degrees of freedom being the rear frame, roll angle ф and the steering angle δ, from upright straight ahead configuration at a forward speed v, looks like this in matrix form :

Fig 4 : LEOM Of A Bicycle

where

M = symmetric mass matrix which gives the kinetic energy of bicycle system at 0 forward speedC1 = damping matrix, proportional to forward speed vK0 = first component of stiffness matrix, which when combined with 'g' yields a symmetric quantity proportional to gravitational acceleration and can be used to calculate changes in potential energyK2 = second component of stiffness matrix, which when combined with the square of forward speed, v, gives a quadratic quantity in forward speed and is due to centrifugal effects.f = applied forces

Consider each of these matrices as packages and the contents of these packages would be specific combinations of the bicycle's design parameters shown in Fig 2 of Part 2. To know what goes where in these matrices, you need to read this paper from Delft.

Because we're analyzing a passive, uncontrolled bicycle, both these moments are taken to be 0. The characteristic equation is then the determinant of the equation in Fig 4 which gives us the eigenvalues of the problem. Eigenvalues are the exponential part of the solution to the differential equations of motion and as said before, help in stability analysis.

Computer Program To Plot Eigenvalues

Solving all this by hand takes pages of tedious work. If we can program these rules into a computer, it can quickly solve the characteristic equation. The input for the program would be all the bicycle's design parameters. The output would be the eigenvalues. We can even tell the computer to plot them for us as a function of forward speed v, for any particular bicycle configuration that we provide.

After checking these boxes, I hit the calculate button on the upper right hand side. The program solves the linearized eigenvalue problem and gives me 4 generalized eigenvalues. It plots these values on the y-axis as a function of forward speed, v on the x-axis.

The above plot tells me something about the stability of this bicycle as a function of speed. To understand what's going on, let us remember the information about eigenvalues in Fig 3 and commit it to mind, or go back and refer to it. Now read the plot in Fig 8 slowly from left to right in the order of increasing speed v. We'll take it piece by piece.

What would you expect from a bike standing still or nearly so? It will simply flop over. How do we know this? The fact that the plot yielded large positive eigenvalues or real numbers tells us right away that this is a very unstable motion. Two positive and negative pairs of roots correspond to both falling and uprighting of the bicycle. One pair corresponds to when the steering is turning toward the lean; the other when it is turning opposite to lean. Since there are no imaginary parts of eigenvalues, it tells us that this capsize motion is non-oscillating, like I mentioned in Part 2.

As the forward speed v is increased from 0.5m/s to slightly more, two real eigenvalues (in blue) become identical, coalesce and form a conjugated pair, which is where oscillatory weave motion actually shows its face. In this oscillation, the bicycle sways about the headed direction.

The positive eigenvalues tend to decrease in magnitude, so the motion is tending towards stability. Eigenvalues with imaginary parts lead to oscillation with increasing frequency, and the rate of increase is rapid at first, but then slows. The bike will weave back and forth one or more times before falling over.

Weave speed is that speed at which weave does not grow or decay, as can be seen by the eigenvalue crossing 0. Hence, weave speed is 4.8539 m/s. It is stable and forms the lower stability range bound for this bicycle. Eigenvalues corresponding to imaginary parts is oscillating motion. Beyond this point, weave is stable until infinity.

This speed range is the stable range for the bicycle, as the eigenvalues corresponding to both weave and capsize have no positive numbers. The bike will weave back and forth, less so each time, and eventually roll straight ahead, although not necessarily in the original direction.

Capsize speed is that speed at which capsize does not grow or decay, as can be seen by the eigenvalue which is at 0. Hence, capsize speed is 4.8539 m/s. It forms the upper stability range bound for this bicycle. Crossing this point gets us over the stable range.

Small positive eigenvalue for capsize gets it into unstable mode. Eigenvalues with imaginary parts, but whose real component is much smaller than the positive eigenvalue overwhelms oscillations. The bike slowly leans farther and farther to one side, without oscillation, until it finally falls over.

Thus, stable speed range for an uncontrolled Litespeed Ultimate is between 10.858-15.880 mph, but for all practical purposes, we could say it becomes easily balanced above 2 m/s. One important thing to realize is that capsize instability in these regions is very slow and thus can be easily corrected by a controlling rider. Also note that all this time, we have deliberately avoided talking about wobble. Wobble is complex and cannot be described without analyzing tire dynamics. This is an on-going study in bicycle dynamics.

In the final series to come shortly, I'll show you a cool peice of literature one of my readers has authored from which we might be able to study how changing the parameters in bicycle design affects the modes of bicycle motion. Now you can all take a deep sigh and have a good weekend.

I haven't studied what you wrote in detail [apologies, I'm frantic with various obligations] but it looks like a very nice and accurate description of our work. Congratulations for 'getting it', and presenting it so nicely.

But that said, I want to talk a little about the limitations. JBike6 is all about inherent stability, and the model is clearly a little unrealistic -- no hands, a perfectly rigid rider, and none of the nasty frictional properties of tires. I don't want to say the modeling is wrong, but there remain some questions to be answered before interpreting the results in terms of rideability. Actually, the equations were validated quite well [by Kooijman and Schwab] for a riderless bike, which also meant the tires were not compressed much [so they had a short contact patch and could possibly act more like ideal rigid wheels].

In terms of ridden bikes, I surmise that JBike6 might apply best to recumbents (where the rider is secured to a seat) with extremely hard tires, ridden no-hands of course.

My personal view, with which Schwab and others may not fully agree, is that true bicycling by a human involves (a) the pressure and mass of the lower arms attached to the handlebars (b) the rider being free to tilt and rotate at the waist, where a system of muscles and sensors are used to keep the rider more or less lined up with the frame (c) the aforementioned tires (d) a control strategy for adjusting bicycle+rider lean for balancing and turning (e) a control strategy for making quick maneuvers or recoveries.

If I am right about this, substantial work needs to be done to expand the model, and then to understand and embody strategies for sensing and muscular activation [in both the waist and the arms]. In other words, a long way to go to understand riding (stabilizing and controlling) qualities, as opposed to riderless stability.

Fascinating comments above from (THE) Jim Papadopoulos. Which chime with my own testing (on adjustable geometry bikes, real roads and most important, a variety of people). The human ability to learn and adapt is not to be underestimated. Likewise apart from geometry, riding position, CofG, the person's weight distribution and muscle tone, all along with stuff like stem length, height and bar width.

I would love to have a unifying set of equations, which work for all variables and am fascinated by all these wonderful attempts. These would help with then often un scientific (some would say BS) seen in some magazines about bike A handling characteristics vs Bike B. However I still think testing with a variety of differently experienced riders with still be needed , if only for final verification.

Mark : Thanks for reading. You're hoping something I guess we're all hoping for pretty soon - a unifying equation of motion for a closed loop system that takes into account all the parameters actually seen in real life. Today we have more of rough, final verification that is very subjective and not enough analysis done from a dynamics standpoint.